> Anyway, the bottom line is, why should I buy Mathematica over
> any of the other packages, such as MathCad, Matlab or Maple?
In my observation (and I _haven't_ tried many of the alternative packages,
and don't wish to knock any of their special advantages):
* mma is _extremely_ powerful and broad in scope -- no other package
probably exceeds it in the overall range of things it can do, the special
functions it offers, the special capabilities it has, etc.
* mma runs on _many_ platforms -- virtually anything you can ask for --
which means it's widely used.
* mma is _widely used_ in industry, academics, etc. -- no matter what you
do, many colleagues are probably using it.
* this means _widespread help and support_, development of many special
packages, and that kind of thing is widely available.
* mma has powerful graphing capabilities, of all kinds and varieties (3D,
contour plots, etc.) at least for initial display of results.
* It seems, at least in my experience, to have _few serious bugs_ or
defects (I've seen reviews that point out a few weird bugs).
Those are the reasons that made me decide to concentrate on it rather than
any other program. On the other hand,
* It can be _extremely_ frustrating and maddening to use, partly because
of its generality and power, partly because it's really a massive kluge of
a program, partly because there are always 16 ways, each of them more
complex that the preceding, to do even a simple task.
* It's useless for producing _presentation quality_ graphics (with
attractive labels, etc). You have to export to an external graphics
program to do that.
* It's near-impossible to know or predict how long calculations,
especially numerical calculations, can take to run, or how best to
organize calculations to obtain the fastest evaluation. Changing some
simple thing in a calculation can suddenly increase the running time by
enormous amounts.
* It's very high priced.
* There are lots of other little reasons to hate it, even as you use it.
My 2 cents...